Indian Institute of Technology Ropar
Department of Mathematics
MA201 : Differential Equation(UG Course)
1st semester of academic year 2018-2019
Tutorial Sheet-6
1. Determine the radius of convergence of the given power series
P
∞
x2n
(a) n!
,
n=1
P∞
(x−x0 )n
(b) n
,
n=1
P∞
n!xn
(c) nn
.
n=1
2. Show that x = 0 is an ordinary point of
(x2 − 1)y ′′ + xy ′ − y = 0
but x = 1 is a regular singular point.
3. Discuss the singularities of the equation
x2 y ′′ + xy ′ + (x2 − n2 )y = 0,
at x = 0 and x → ∞.
4. Is x = 0 an ordinary or singular point?
(a) xy ′′ + y ′ + xy = 0
(b) xy ′′ + (sin x)y ′ + x2 y = 0
√
(c) y ′′ + x2 y ′ + ( x)y = 0.
5. For the differential equation
(x − 1)y ′′ + cot(πx)y ′ + cosec2 (πx)y = 0,
which of the following statement is true?
(a) x = 0 is regular and x = 1 is irregular point.
(b) x = 0 is regular and x = 1 is regular point.
(c) both x = 0 and x = 1 are irregular point.
6. For the nonlinear IVP y ′ = x + y 2 , y(0) = 1, find first four nonzero terms of a series
solution y(x) in the following two ways:
P
∞
(a) by setting y = an xn and finding in order a0 , a1 , a2 , . . . using the initial
n=0
condition and substituting the series into the ODE;
(b) by differentiating the ODE repeatedly to obtain y(0), y ′ (0), y ′′ (0), . . . and then
using Taylor’s formula.
(c) Check both the results are same or not?
7. Show that if k is a positive integer 2m, in Hermite’s equation y ′′ − 2xy ′ + ky = 0,
then one of the power series solutions is a polynomial of degree m.
8. Find two independent power series solution about x = 0 of the ODE
(1 − x2 )y ′′ − 2xy ′ + 6y = 0
and determine their radius of converges R. To what extent R is predictable from
the original ODE.
9. Find the power series solution of the initial value problem
(x2 − 1)y ′′ + 3xy ′ + xy = 0, y(2) = 4, y ′ (2) = 6.
10. Consider the differential equation
x3 y ′′ + xy ′ − y = 0
(a) Show that x = 0 is an irregular singular point.
(b) Use the fact that y1 (x) = x is a solution to find a second independent solution
y2 (x).
(c) Show that the solution y2 (x) found in (b) cannot be expressed as a Frobenius
series.
11. Discuss whether two Frobenius series solutions exists or do not exist for the following
equations:
(a) 2x2 y ′′ + x(x + 1)y ′ − (cos x)y = 0
(b) x4 y ′′ − (x2 sin x)y ′ + 2(1 − cos x)y = 0.
12. Use the Frobenius method to find two linearly independent series solution near x = 0
9x(1 − x)y ′′ − 12y ′ + 4y = 0.
13. Find two independent Frobenius series solutions near x = 0
(1 − x2 )y ′′ − xy ′ + 4y = 0.
14. Find the Frobenius series solution of Bessel’s equation of first order,
x2 y ′′ + xy ′ + (x2 − 1)y = 0
near x = 0.
15. Use the Frobenius method to find two linearly independent series solution near x = 0
(x − x2 )y ′′ + (1 − 5x)y ′ − 4y = 0.
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